On the End-to-end Call Acceptance and the Possibility of Deterministic QoS Guarantees in Ad hoc Wireless Networks

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1 On the End-to-end Call Aeptane and the Possblty of Determnst QoS Guarantees n Ad ho Wreless Networks S. Srram T. heemarjuna Reddy Dept. of Computer Sene Dept. of Computer Sene and Engneerng Unversty of Calforna, erkeley, CA 947 Indan Insttute of Tehnology Madras, Inda srram s@s.berkeley.edu arjun@s.tm.ernet.n. S. Manoj C. Sva Ram Murthy CalIT, Jaobs Shool of Engneerng Dept. of Computer Sene and Engneerng Unversty of Calforna, San Dego, CA 99 Indan Insttute of Tehnology Madras, Inda bsmanoj@usd.edu murthy@tm.a.n ASTRACT The ssue of provdng Qualty of Serve (QoS) guarantees n an Ad ho wreless network s a very hallengng problem. In ths paper, we make the followng ontrbutons: () analytally derve bounds for the end-to-end all aeptane rate usng exstng queueng theory methods, () study the mpat of the routng sheme on the end-to-end all aeptane rate, and () propose a dfferentated serves sheme for determnstally provdng QoS guarantees. Unlke exstng studes whh analyze the transport apaty, we fous on the end-to-end all aeptane. The framework that we assume s that of a TDMA-based Ad ho wreless network. The routng sheme employed nfluenes the end-to-end all aeptane of the network. The metrs that we onsder are the all aeptane probablty and the system saturaton probablty (.e., the probablty that the network s n a state n whh every new all s rejeted). We derve general bounds on the all aeptane and the system saturaton for the ase of dfferentated-lasses of users n the network. These bounds ndate the number of alls of the hghest prorty lass that an be admtted nto the network. Smulaton studes were arred out to study the effet of load, hopount, and the nfluene of the routng protool on the all aeptane. The nrease n the all aeptane rate wth the ntroduton of load-balanng hghlghts the mportane of load-balanng n enhanng the system performane. From these studes, we arrve at the followng results: () load-balanng leads to sgnfant Ths work was supported by Nautx Tehnologes Inda Prvate Lmted, Chenna, Inda, Department of Sene and Tehnology, New Delh, Inda, and Mrosoft Researh Unversty Relatons Inda (Award Number 75). Author for orrespondene Permsson to make dgtal or hard opes of all or part of ths work for personal or lassroom use s granted wthout fee provded that opes are not made or dstrbuted for proft or ommeral advantage and that opes bear ths note and the full taton on the frst page. To opy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror spef permsson and/or a fee. MobHo 5, May 5 7, 5, Urbana-Champagn, Illnos, USA. Copyrght 5 ACM /5/5 $5.. mprovement n the end-to-end all aeptane rate, and s an mportant fator n attanng the maxmum end-to-end all aeptane rate n a gven network and () t s ndeed possble to provde determnst QoS guarantees for a desgnated set of nodes whh are haraterzed by determnst guarantee lmt. Categores and Subjet Desrptors: C.. [Network Arhteture and Desgn] Wreless ommunaton; C.. [Network Protools] Routng protools; C.4 [Performane of Systems] Modelng tehnques, Performane attrbutes General Terms: Algorthms, Theory, Performane eywords: Ad ho wreless networks, QoS guarantees. QoS routng, TDMA, all aeptane rate, load-balanng, Markov proess. INTRODUCTION An Ad ho wreless network s a olleton of moble nodes that an ommunate over rado wthout any pre-exstng nfrastruture. Two nodes an ommunate dretly wth eah other f eah les n the transmsson range of the other. Two nodes that annot dretly ommunate an do so n a mult-hop manner n whh the other nodes funton as routers. Suh networks are used n mltary nstallatons and n emergeny stuatons as they permt the establshment of a ommunaton network at very short note. However, these networks are lmted by onstrants n ther bandwdth and power onsumpton. Wth ther wdespread deployment, Ad ho networks now need to support applatons that generate real-tme traff. Suh traff requres the network to provde guarantees on the QoS of the onneton. The mportant aspets n the proess of provdng suh guarantees are the routng protools that establsh paths that an satsfy the QoS requrements and the reservaton mehansms that reserve the neessary resoures along the path. A problem of onsderable nterest n ths regard s that of theoretally estmatng the nature of the guarantees that an be provded by a QoS sheme. These estmates on the parameters of QoS routng protools gve us an dea of the maxmum guarantees that an be provded, and allow us to gauge how far the exstng shemes are from the deal lmt. In ths work, we onsder the problem of QoS routng n a TDMAbased Ad ho wreless network, where the QoS onstrant on the alls s that of bandwdth. Our fous s the end-to-end all aep- 69

2 tane rate whh s a measure of the number of alls that an be admtted nto the network. The alls arrvng n the network belong to dfferent lasses based on whh the requrements of the alls are prortzed. Thus, the parameters that we fous on are: the all aeptane probablty and the system saturaton probablty. The varaton of these parameters enables us to answer questons suh as ) What are the maxmum number of hgh-prorty alls that an be sustaned n the network at a gven load?, ) What s the lkelhood that the network enters a state where no more new alls an be aepted?, ) What s the effet of the routng protool on the all aeptane?, 4) How lose to the theoretal lmt do the routng protools approah? Then, we address the problem of ensurng determnst all aeptane for a ertan sub-set of the alls. We estmate the determnst guarantee lmt whh s a mobltyndependent measure of the number of hgh-prorty alls that an be admtted nto the network. We also determne the all aeptane probablty for the lasses for whh determnst guarantee annot be provded. In ths work, we model the network at the level of the transmsson range of eah node. The range of a node s analyzed as a Markov proess where the alls are the enttes to be served. The reservaton of slots for the all n the transmsson range onsttutes the serve of the all. The modelng of a wreless network as a olleton of Markov proesses s unque n that, due to the loal broadast nature of the hannel, the reservaton of slots n the transmsson range of a node affets the status of the slots n the neghborng regons. Capturng ths property of wreless networks s essental to model the haratersts of the network aurately. Suh a modelng must also be able to reflet the haratersts of the routng protool used. We begn by analyzng a general ase of a network that an support multple-lasses of alls where preempton of alls does not exst. We then provde a losed-form estmate of the all aeptane probablty and the saturaton probablty for the ase of a sngle-lass of users and dsuss the probabltes for the hghest-prorty lass n the preemptve ase. We ompare the all aeptane probabltes of shortest-path routng and two routng protools that attempt load-balanng. Fnally, we estmate the determnst guarantee lmt. The rest of ths paper s organzed as follows: Seton brefs the related work n ths area, Seton desrbes our work, Seton 4 dsusses the detals of the smulaton, and Seton 5 presents the smulaton results. Fnally Seton 6 onludes the paper.. RELATED WOR In ther semnal work[], Gupta and umar ntrodued a random network model for studyng throughput salng n a fxed wreless network. They showed that even under optmal ondtons, the transport apaty (bt-dstane produt that an be transmtted over the network) of the network s θ( n) bt-meters/s, where n s the number of nodes present n the network, for the protool model onsdered. In [] an nformaton-theoret sheme was onstruted for obtanng an ahevable rate regon n a network of arbtrary sze and topology. The proposed sheme allows a feasble transport apaty of θ(n) bt-meters/s n a spef wreless network of nodes loated n a regon of unt area, as ompared to θ( n) bt-meters/s obtaned n [] for less sophstated reever operaton. In [], the authors showed that by allowng nodes to move, the throughput salng hanges dramatally. They showed that f node moton s ndependent aross nodes and has a unform statonary dstrbuton, a onstant throughput salng (θ()) per soure - destnaton par s feasble. In [4], the authors studed the transport apaty of an Ad ho wreless network overlad wth an nfnte apaty nfrastruture network. Whle the prevous studes analyze the transport apaty of Ad ho networks, n ths work our fous s on the end-to-end all aeptane rate whh s a measure of the number of alls wth endto-end bandwdth reservaton that an be supported by the network. The prevous studes on transport apaty study how t sales wth the number of nodes. We study the dependene of end-to-end all aeptane rate on the network load and the routng protool. Our work attempts to arrve at generalzed bounds that an be used to analyze routng protools. A number of QoS routng protools have been proposed for Ad ho wreless networks (for more detals, refer [5], [6], [7], and [8]). The framework that we assume s that of a TDMA-based network. The routng sheme employed nfluenes the end-to-end all aeptane rate of the network. In ths paper, we nvestgate the end-to-end all aeptane rate and the nfluene of shortest-path routng and load-balaned routng protools on t.. OUR WOR We onsder an Ad ho wreless network omprsng N nodes unformly dstrbuted at random n a terran of area A. The transmsson range of eah node s R. We assume the presene of a slotted TDMA mehansm at the MAC layer. The total number of slots avalable n the network s. However, t s possble to reuse the slots spatally dependng on the nterferene pattern of the nodes. Ths s the key dea that s used n dervng the bounds. The bandwdth of a all s measured n terms of the number of slots used for transmsson. A all s setup by reservng slots along the path of the all. A node may ether transmt or reeve n a partular slot (a node s sad to reeve n a partular slot f any of ts neghbors s transmttng n that slot). A slot s sad to be free at a node j, ( j N) f t s nether transmttng nor reevng durng that slot. For a node j to transmt n a partular slot, the slot must be free at j and none of the neghbors of j must be reevng n that slot. For a node j to reeve n a partular slot, the slot must be free at j. Ths defnton permts node 6 to transmt to node 5 n the same slot as the one used by node to transmt to node n Fgure, provded nodes and do not hear 6. On the other hand, n the sender s range, node 4 must use a dfferent slot to transmt to node beause node hears the transmsson by node. 4 Fgure : An example of possble transmssons.. System Model Consder a network of NW = {,,N} of N nodes that an support lasses of alls where lass alls have a hgher prorty 5 6 7

3 than lass j ( <j ) alls. We would lke an estmate of how many alls of a partular lass an be supported. Ths mples that we an defntely support suh a number of lass alls where lass s the hghest prorty lass. If all the avalable slots are ouped by alls of varous lasses upon arrval of a lass j all, one or more alls of lower prorty lasses an be preempted based on the bandwdth requrement of lass j all that has arrved to ensure that the arrved all be aepted. Thus, we would lke to provde a guarantee on the number of alls of a partular lass that an be aepted. A4 # A # A # A A5 R A # # A A # #4 A4 A5 #5 A6 #x slot number x Fgure : An example senaro. Assumptons made for the analyss are the followng: Calls of a partular lass-k arrve at eah node dstrbuted aordng to a Posson proess of mean λ k. The duraton of a all s exponentally dstrbuted wth mean duraton µ k. We assume that the alls of all lasses have equal bandwdth requrements: eah all requres a sngle slot. We do not take node moblty nto aount n the estmaton of all aeptane and system saturaton. (However, the determnst guarantee lmt s ndependent of moblty.) We assume that the routng algorthm s suh that for any path found by the algorthm, the number of nodes on the path that le wthn the transmsson range of any node on the path (nlusve of the node tself) s not greater than some onstant. In the absene of suh an assumpton, t s possble to onstrut a senaro (Fgure ) where a sngle all needs to use all the slots n the system. In Fgure, eah of the nodes on the path s n the transmsson range of the other nodes. So the use of a slot for transmsson by one of the nodes mples that the slot annot be re-used by the other nodes on the path. Thus, f node A transmts to node A on slot #, slot # annot be used by any of the other nodes to transmt to ther downstream nodes. If we were to onsder a P -hop path wth the nodes n the onfguraton gven n Fgure, the number of slots used would be P. Hene, t would be dffult to provde a bound on the number of alls that an be admtted. Ths property s satsfed wth =for protools that ensure that f a path s to be set up from A to A, the path used s the lnk (A,A ) rather than lnks (A, A ) and (A, A ), Fgure : An example to dstngush free slots at a node and free slots n a regon. where A, A, and A are nodes suh that eah an lsten to the other two. Ths an be done by usng an approprate forwardng of the route request pakets n whh a node drops all exept the frst route request that t reeves.. Theoretal Analyss Intally, we assume that all preempton does not our. We derve upper and lower bounds on the all aeptane probablty for the ase of sngle-hop and mult-hop alls, respetvely. Consder a node j and the regon spanned by ts transmsson range R(j). Any all passng through R(j) uses up some number of slots. The number of slots used up n the regon R(j) depends on the number of alls orgnated from node j, the number of alls from any of the neghbors of node j, and the number of alls that orgnate from outsde R(j) and are routed through R(j). A slot s sad to be free n R(j) f no nodes n R(j) are ether transmttng or reevng n that slot (.e., slot s free at all nodes n regon R(j)). In Fgure, node A transmts to node A on slot. Node A 4 transmts to node A on slot. Node A 5 transmts to node A on slot. If the network had a total of 5 slots, the free slots at node A would be {,, 4, 5} whle the free slots n the regon R(A ) would be {4, 5}. We an thus vew R(j) as a server of slots for whh the alls ontend. Although the dstrbuton of all arrvals of a partular lass at eah node s known to be Posson, the dstrbuton of alls arrvng at R(j) s not Posson due to the splttng of the Posson streams (Consder alls arrvng at a node based on a Posson proess of mean λ. Assume that the node has to forward the all along one of two lnks. If the node forwards alls n a nonrandom manner, the arrval of alls at the downstream node wll no longer be Posson). We make use of lenrok s Independene Assumpton, aordng to whh, for moderately heavy all arrval at eah node, the net all arrval at the regon R(j) an be regarded as Posson. Thus, alls of a lass-k arrve at R(j) aordng to a Posson dstrbuton wth mean: =N λ k (j) = f k (, j)λ k = where f k (, j) s the fraton of lass-k alls orgnatng n node that pass through the regon R(j). Ths an be rewrtten as: λ k (j) =( f k (, j)+ N(j) +)λ k () / R(j) 7

4 (n, n,, n,, n ) n µ t (n +, n,, n,, n ) (n, n +,, n,, n ) (n +)µ t (n, n,, n,, n ) µ t n (n, n,, n,, n ) (n +)µ t (n +) µ t µ t n (n, n,, n +,, n ) (n, n,, n,, n ) n µ t (n +)µ t (n, n,, n,, n ) (n, n,, n,, n +) Fgure 4: The transtons nto and out of one of the states of the Markov proess representng the regon R(j). For the state (n,n,,n ), n >,n >,,n >. where N(j) denotes the set of nodes n the transmsson range of node j. The parameter f k (, j) s dependent on the routng protool. For a protool suh as shortest-path routng, whh leads to heavy loads n the enter of the network, f k (, j) would be hgh for nodes j ( j N) loated near the enter. For protools that mplement load-balanng, the value of f k (, j) should be farly unform aross the nodes. The state of the system R(j) s gven by the number of alls of eah lass beng served (eah of whh uses up some of the slots of R(j)) byr(j). We thus model R(j) as a -dmensonal dsretetme Markov proess (t) =(n,,n ), where n k denotes the number of lass-k alls beng served by R(j) at tme t []. We denote: P ((n,,n ) (n,,n )) = P ((t + t) = (n,,n ) (t) =(n,,n )) as the probablty that the system R(j) s n the state (n,,n ) at tme t + t gven t s n the state (n,,n ) at tme t. P ((n,,n k +,,n ) (n,,n k,,n )) = λ k (j) t () P ((n,,n k,,n ) (n,,n k,,n )) = n k µ k t, n k > () The state-transton dagram representng the transtons nto and out of one of the states of the Markov proess s shown n Fgure 4. The Markov proess has a unque steady-state probablty dstrbuton []. Usng Equatons () and () along wth the normalzaton of probabltes, we an alulate the probablty that the In the most general ase of a model orrespondng to lasses of alls, the Markov proess has `+ states. Ths s not a problem for the urrent analyss sne the transtons between the states are restrted: every state has at most neghborng states, and the proesses assoated wth any gven regons are deoupled. Further, we are nterested n only the steady state of the proess and not n the paths traversed. The state-exploson needs to be takled for an analyss that onsders oupled proesses or preemptve alls: the nterested reader may refer [9] and []. For the ase of preempton, the system an move between ertan other states. Correspondng to the ase of preempton of a lass- all by a lass- all, the system an move from the state (n,n,,n ) to (n +,n,,n ),n. system s n a partular state (n,,n ) as: P ((n,,n )) = k (j) n k G(j) k= where k (j) = λ k(j) and µ k G(j) = n +,,+n k= k (j) n k (4) s a normalzaton fator. We would now lke to extend ths Markov proess to dstngush between alls that termnate n a node n R(j) (all them type-u alls) and those that do not (type-v alls). Let us say that a fraton f of the alls termnate n some node n R(j). If the destnaton were to be hosen randomly, then f = N(j) +. The state of the N system s now gven by: (n,u,n,v,n,u,n,v,,n,u,n,v ) where n k,u s the number of lass-k alls that are type-u alls n R(j) and n k,v s the number of lass-k alls that are type-v alls. The probablty that the system s n a state (n,u,n,v,n,u,n,v,,n,u,n,v ) s: P ((n,u,n,v,,n,u,n,v )) = k,u (j) n k,u k,v (j) n k,v (5) E(j) n k,u! n k,v! k= where k,u (j) = fλ k(j) µ k, k,v (j) = ( f)λ k(j) µ k, and E(j) = n,u,n,v,,n,u,n,v k= k,u (j) n k,u n k,u! k,v (j) n k,v n k,v! a normalzaton fator. The probablty that the system s n a state (n,v,n,v,,n,v ) (a state n whh there are n,v lass- type-v alls, n,v lass- type-v alls, and so on) s: P ((n,v,,n,v )) = k,v (j) n k,v H(j) n k,v! k= (6) where H(j) = s a normalzaton fator. n,v +,,+n,v k= k,v (j) n k,v n k,v! s 7

5 C C C j C5 Fgure 5: In the regon R(j), C and C4 are type-u alls; C, C, and C5 are type-v alls. For eah type-v all, we see that at least one slot that has not been used so far n R(j) must be used. For the type-u alls, slot reuse s possble n some ases. Theoretal upper bound for probablty of all aeptane We now derve an upper bound on the probablty of all aeptane for the ases of sngle-hop and mult-hop alls. Sngle-hop ase: Consder a sngle-hop all from node j to ts neghbor node l. For the all to be aepted, at least one slot must be free n the regon R(j). Thus P A (probablty of a sngle-hop all s aepted) s: P A = P (Number of free slots ) = P (Number of used slots ) where s the total number of slots n the system. From Lemma : P A P (Number of type V alls ) P (Number of type V alls > ) P (Number of type V alls = ) P A n,v +n,v ++n,v = k,v (j) n k,v H(j) n k,v! k= (7) For the ase of a sngle-lass of alls, Equaton (7) redues to.. Call Aeptane Probablty In ths seton, we are gong to derve the all aeptane probablty of both sngle-hop and mult-hop ases for a non-preemptve system (a system where the aepted alls are not dropped for a new all). LEMMA. P (Number of used slots n a regon R(j) x) P (Number of type V alls n R(j) x), where x N. Proof: For every type-v all, at least one unque (untl then unused) free slot n the regon R(j) must be used (see Fgure 5). Thus: Number of type V alls n R(j) >x Number of used slots n R(j) >x and Number of used slots n R(j) x Number of type V alls n R(j) x Hene P (Number of used slots n R(j) x) P (Number of type V alls n R(j) x) LEMMA. P (Number of alls n a regon R(j) x) P (Number of used slots n a regon R(j) x), where x N and s the routng algorthm dependent onstant fator that denotes the maxmum number of nodes on a path that le wthn the transmsson range of any node on the path. Proof: Number of alls n R(j) x Number of used slots n R(j) x Hene P (Number of alls n R(j) x) P (Number of used slots n R(j) x) P A,V (j) (8) H(j)! Mult-hop ase: We set the onstant =. Consder a (M )- hop all (M ) setup along the nodes (p,,p M). When a slot s reserved for transmsson between p and p, the total number of free slots at R(p ) dereases by (sne the slot annot be used for transmsson from p to p ). Thus, the total number of slots avalable at R(p ) an be onsdered as. Call ths modfed regon R (p ). When slots have been reserved between p and p, and between p and p, the number of free slots at R(p ) dereases by so that the total number of slots at R(p ) an be regarded as. Call ths modfed regon R (p ). The number of slots, for the regons R(p ),,R(p (M ) ), s thus effetvely, (sne =). (Thus, aordng to ths notaton, a regon R (j) has one fewer slot, whle R (j) has two fewer slots). A mult-hop all setup for M =5s shown n Fgure 6. A all s suessfully forwarded n regon R(j) f slots an be found n R(j) so that the all havng arrved at node j s forwarded to ts next hop n the path. For the all to be aepted, t must frst be suessfully forwarded n the regon R(p ), must then be suessfully forwarded through eah of the regons R (p ),R (p ),, P P P P4 Slot # Slot # P5 Slot # Slot # Fgure 6: Mult-hop all setup. R(P ) needs slot # to be free. R(P ) now annot use slot # and requres slot # (some other slot) to be free. R(P ) annot use slots # and #, and requres slot # (any other slot) to be free. R(P 4) an transmt n slot # f t s free. 7

6 R (p M ). A neessary and suffent ondton for suessful forwardng s the presene of at least one free slot n eah of the ntermedate regons. Denote: Suessful forwardng of all n R(j) as SF of all n R(j). Thus P A s gven by: P A = P (SF of all n R(p )) P (SF of all n R (p ) SF of all n R(p )) P (SF of all n R (p ) SF of all n R (p )) P (SF of all n R (p M ) SF of all n R (p M )) k,v (p ) n k,v C P A n,v +n,v ++n,v = H(p ) k= n k,v! k,v (p ) n k,v n,v +n,v ++n,v = H A (p ) k= n k,v! k,v (p ) n k,v n,v +n,v ++n,v = H A (p ) k= n k,v! k,v (p M ) n k,v n,v +n,v ++n,v = H A (p M ) k= n k,v! k,v (p M ) n k,v n,v +n,v ++n,v = H A () (p M ) k= n k,v! where H (j) = H (j) = P A n,v +,,+n,v k= n,v +,,+n,v k= k,v (j) n k,v n k,v! k,v (j) n k,v n k,v! For the ase of a sngle-lass of alls, Equaton () redues to»,v (p )»»»» H(p ) H (p ) H (p ) H (p M ) H (p M )!,V (p ) ( )!,V (p ) ( )!,V (p M ) ( )!,V (p M ) ( )!. and () The RHS (Rght Hand Sde) of Equatons (8) and () are hard to solve for n a losed-form. For moderate-to-heavy traff, > and the nequalty remans vald f we replae,v (p j), j M by Max,V (the maxmum value of,v (p j) aross all the regons). Denotng the RHS as PA Max : P Max A = H,V Max! for sngle hop alls () " # " # PA Max = Max,V Max,V H! H ( )! " # Max M,V for mult hop alls () P A = P (No. of free slots n R(p ) ) H ( )! P (No. of free slots n R (p ) SF of all n R(p )) b= P (No. of free slots n R (p ) SF of all n R Max b (p,v )) where H =, b! b= P (No. of free slots n R (p M ) SF of all n R b= (p M ))(9) H Max b b=,v =, and H Max b,v =. b! b! From Lemma : b= b= P A P (No. of type V alls n R(p ) ) P (No. of type V alls n R (p ) ) P (No. of type V alls n R (p ) ) Theoretal lower bound for probablty of all aeptane In ths seton, we derve lower bounds on the probablty of all aeptane for the ase of sngle-hop and mult-hop alls. Sngle-hop ase: For the sngle-hop ase, a all from node j to P (No. of type V alls n R (p M ) ) ts neghbor node l s aepted f there s at least one free slot n the regon R(j). From our assumpton about the fat that the routng protool satsfes the property that at most nodes on the path an hear any other node on the path, we have for a gven number of alls n the regon R(j) P A = P (Number of free slots ) = P (Number of used slots ) Usng Lemma P A P (Number of alls G(j) For a sngle-lass of alls n +n ++n ) k= k (j) n k (4) (5) = (j) P A (6) G(j) = Mult-hop ase: Consder the attempt to setup an (M )-hop all (M ) along the nodes(p,,p M). The probablty of all aeptane s gven by Equaton (9). From Equaton (9) and Lemma P A P (Number of alls n R(p ) ) P (Number of alls n R (p ) ) P (Number of alls n R (p ) ) P (Number of alls n R (p M ) ) 74

7 P A G(p ) G (p ) where G (j) = G (j) = G (p M ) G (p M ) n +n ++n n +n ++n k= k= n +n ++n n +n ++n n ++n k= n ++n k= k (j) n k. k (j) n k k (p ) n k k (p ) n k k= k= and k (p M ) n k k (p M ) n k For the sngle-lass ase: P A = (p ) G(p ) = = (p ) G (p ) = = (p M ) G (p M ) = = (p M ) G (p M ) = (8) Usng the same approxmatons as n Equatons () and (), we an determne the mnmum value of the aeptane probablty PA Mn : P Mn A P Mn A = = G 6 4 G 6 4 G = where G = G = = = Max = = = = = = Max. = Mn, G = Mn Mn for sngle hop alls (9) G 7 5 = = M Max = = Mn 7 5 for multhop alls (), and.. System Saturaton Probablty For the ase of a sngle-lass of alls, the probablty that the network s saturated.e., no further alls an be aepted s gven by P Sat. If the number of type-v alls n a regon s, then ths (7) would requre at least slots to be used, and no further alls an be aepted. P (Saturaton n R(j)) = P ( slots are used) P (Saturaton n R(j)) P (Number of type V alls at R(j) =) P Sat H(j) =N,V (j)! H() =,V ()! P Sat [ Max N,V ] H!.. A Summary of the Results () () () The Equatons (9), (), (), and () suggest that the all aeptane dereases wth system load, ths derease beng rapd at hgh loads. For an nomng all s hanes of aeptane to be maxmzed, Equatons (7) and () suggest that the mnmum (j) aross the network be maxmzed: ths suggests that loadbalanng would help mprove the aeptane rate. If all the nodes are wthn the transmsson range of one another (all ommunaton s sngle-hop), then the upper and lower bounds (Equatons (9) and ()) onverge wth k (j) =N λ k. µ k To ensure that the all aeptane s always above a ertan threshold rrespetve of the load, Equatons (9) and () ndate that the network must be well-provsoned.e., must be suffently hgh. As boundary ases, the followng are seen to hold for the all aeptane rates: as the number of slots nreases, t tends to unty. As the all duraton nreases, t approahes zero...4 The Case of Preempton The analyss so far has been done under the assumpton that hgh-prorty alls annot preempt lower-prorty ones. However, a realst senaro may requre that hgh-prorty alls are ensured hgh probablty of all aeptane. Ths may requre ntroduton of preempton nto the system. The analyss of the steady-state probabltes of a preemptve Markov proess s a dffult problem. The statonary dstrbuton of the hghest prorty alls an be easly obtaned sne these alls effetvely gnore the presene of other low-prorty alls. Thus, the statonary dstrbuton of the lass- alls s the same as that of the sngle-lass system gven n Equatons (9), (), (), and ()...5 The Falure of Shortest-path Routng The analyss tells us that the parameters: the all aeptane probablty and the system saturaton probablty depend on the load on the network, the hopount of the path, and the routng protool. We frst look at the performane of shortest-path routng relatve to the theoretal guarantees. The routng protool s related to the all aeptane and the system saturaton probablty through the fator f k (, j) spefed n Equaton (). Shortest-path routng: Shortest-path routng omputes the shortestpath between the soure and the destnaton where the dstane 75

8 5 Normalzed average f for Shortest path routng for.4 alls per seond PA vs Maxmum Shortest-path Mnmum 4.8 Normalzed average f PA.6.4. Rng Rng Rng Rng Rng number Fgure 7: The normalzed average fraton of alls beng routed to a node wth nreasng dstane from the enter for shortestpath routng. The arrval rate s.4 alls per seond at a node. 5 Normalzed average f for Shortest path routng for alls per seond Fgure 9: Call aeptane probablty of sngle-hop alls usng shortest-path routng vs varyng load..8 PA vs Maxmum Shortest-path Mnmum 4.6 Normalzed average f PA Rng Rng Rng number Rng Rng Fgure : Call aeptane probablty of -hop alls usng shortest-path routng vs varyng load. Fgure 8: The normalzed average fraton of alls beng routed to a node wth nreasng dstane from the enter for shortestpath routng. The arrval rate s alls per seond at a node. refers to the Euldean dstane between the soure and the destnaton. In a hghly dense network, the authors of [] proved that the average path length obtaned when shortest-path routng s employed s.95r where R s the radus of the network. Ths leads to heaver load at the enter regon of the network. We smulate shortest-path routng and measure the all aeptane rate. The Fgures 7 and 8 ndate the loadng of the enter of the network, and dereasng load away from the enter where the rng an be regarded as a unt of dstane from the enter (refer Seton 4 for more detals). The Fgures 9,, and show that the shortestpath routng has a all aeptane rate muh below the theoretal lmt. Note that even n Fgure 9, the system has several alls wth varyng hops, whh would be the ase n a realst senaro. The results shown n Fgures 9,, and are got by measurng the aeptanes for sngle-hop, -hop, and -hop alls, respetvely. The reason that shortest-path routng performs badly s due to the fat that a majorty of the alls are routed through the enter of the network resultng n a hgh load n the enter. Ths problem suggests the use of load-balanng to allevate the formaton of hotspots and to nrease the all aeptane.. Load-balanng We onsder the followng strateges for load-balanng: Rng-based routng: Rng-based routng [] transfers the load from the enter to the perphery of the network. The sheme makes use of heursts to balane the load. We defne the followng terms: The enter node or enter of a network, C, s the node for whh, max x (HC(C, x)) mn y (max z (HC(y,z))) for all nodes x, y, and z n the network. Here HC(a, b) denotes the hopount of the shortest path from node a to node b. Eah node n the network belongs to a Rng denoted by Rng (r,r +). ARng s an magnary dvson of the network nto onentr rngs about the enter of the network. The thkness of the rng s gven by r + r. A node that belongs to Rng les at a dstane n (r,r +) from the enter of the network. The load balanng heurst that we use s a Preferred Outer Rng routng Sheme (PORS) []. In ths strategy, traff 76

9 PA PA vs Maxmum Shortest-path Mnmum Table : Parameters used n the smulaton Parameter Value Number of nodes 5 Number of slots Terran area m m Transmsson range m Average all duraton s Smulaton duraton s Number of seeds Fgure : Call aeptane probablty of -hop alls usng shortest-path routng vs varyng load..8 PA vs Maxmum W PORS Shortest-path Mnmum.6 generated n a node n Rng and destned for a node n Rng j must not go beyond the rngs enlosed by Rng and Rng j. Further, the pakets must be preferentally routed through the outer of the two rngs. Thus, for nodes belongng to the same rng, pakets must be preferentally transferred n the same rng. For nodes belongng to dfferent rngs, all angular transmssons must preferentally take plae n the outer of the two rngs whle the radal transmssons transfer pakets aross the rngs. Thus, PORS affets the hopount whle at the same tme movng most of the load away from the enter. andwdth-lmted routng: andwdth-lmted routng s a more dret form of load-balanng that uses an estmate or measurement of the avalable bandwdth to selet a path. It dffers from the two prevous methods (shortest-path routng and PORS) n that t s dynam: onstantly adaptng to hanges n the network state. There are two opposng metrs that suh a sheme attempts to reonle. It tres to hoose paths wth the hghest avalable bandwdth. These paths, usually, tend to be longer than the shortest path. As a result, the avalable bandwdth of the path, whh s the mnmum of the avalable on the onsttuent lnks, s more lkely to derease. The sheme that we use s based on the Shortest-dst (P, n) studes n []. Shortest-dst (P, n) heurst fnds a path P wth the shortest dstane k dst(p, n) = r n = where r,,r k are the max-mn far rates of lnks on the path P wth k hops. We use a varant of ths heurst. The weght for the lnk (u, v) s weghted by (u, v) n where (u, v) s the estmated bandwdth of the lnk, and n s a weghtng fator. We smply estmate ths as the mnmum of the number of free slots at nodes u and v. The ntuton behnd ths heurst s that when the lnks are weghted thus, shortest-path routng wll selet a path that mnmzes P =k d = n where k s the number of hops, d s the Euldean dstane of the th hop, and s the estmated bandwdth of the lnk traversed on the th hop. Ths heurst tends PA Fgure : Varaton of Call Aeptane vs for Sngle-hop alls. to selet lnks wth hgh avalable estmated bandwdth that would also form a short path to the destnaton. We set the exponent n to for our experments. 4. SIMULATION STUDIES To study the atual behavor of the parameters of nterest, we bult an Ad ho wreless network smulator n C++. The network s TDMA-based. Reservaton nvolves two steps: fndng a path usng one of the routng protools dsussed and reservng slots along the path. Slot alloaton for a partular all s done n a greedy manner. If at any ntermedate node, the number of free slots s found to be nadequate, the all s rejeted. Calls are generated at eah node aordng to a Posson proess and the aepted alls have an exponentally dstrbuted all duraton. The nodes are not moble. The parameters of the smulaton are spefed n Table. The smulatons are run wth seeds: eah run generates a random topology. In eah run, alls are generated randomly aordng to a Posson dstrbuton wth an exponentally dstrbuted duraton. We ran the smulatons for a duraton of s. For the smulaton studes, we vary the load by varyng the all arrval rate at eah node. We ompare the all aeptane probabltes for varyng values of the rato = (Average Call Arrval Rate Average Call Duraton). In order to ompare the theoretal values and the expermental results, we need to translate the value to the Max,V value. Thus, we also measure the average fraton of alls that pass through a regon. Ths fator s an ndaton of the nature of the routng protool used. We then measure the all aeptane of alls based on ther hopount for dfferent routng protools and ompare wth the theoretal lmts. Results presented n ths paper onform to 95% onfdene ntervals. 77

10 PA PA vs Maxmum W PORS Shortest-path Mnmum Normalzed average f 5 4 Normalzed average f for.4 alls per seond Shortest path PORS W Fgure : Varaton of Call Aeptane vs for -hop alls. Rng Rng Rng Rng Rng number Fgure 5: The normalzed average fraton of alls beng routed to a node wth nreasng dstane from the enter. The arrval rate s.4 alls per seond at a node. PA PA vs Maxmum W PORS Shortest-path Mnmum Normalzed average f 5 4 Normalzed average f for alls per seond Shortest path PORS W Fgure 4: Varaton of Call Aeptane vs for -hop alls. 5. SIMULATION RESULTS 5. Call Aeptane Probablty We have ompared the probablty of all aeptane of shortestpath routng, andwdth-lmted routng (W), PORS, and the theoretal bounds at dfferent values of load (n terms of the rato ).We have also studed the aeptane probablty for hopount values of,, and (Fgures,, and 4). In all the results, the all aeptane probablty value dereases wth an nrease n the network load, as expeted. Further, the urves deptng the all aeptane probablty values of shortest-path routng, W, and PORS le wthn the regon surrounded by PA Max and PA Mn. PORS performs only margnally better than shortest-path routng (and n fat worse for sngle-hop alls) whle W performs sgnfantly better. PORS attempts to load balane mpltly by routng alls to the perphery: ths may not be the most effetve strategy beause nodes n one rng an nterfere wth those n the other rngs. Also t does not take nto aount the fat that a longer path would result n more resoures beng onsumed affetng the aeptane rate of alls n the future. Ths s probably the reason why the sngle-hop alls have a lower aeptane rate n PORS. W, by usng an explt bandwdth-based load-balanng s evdently more effetve. To brng out the dfferene n the performane of the three routng algorthms, we ompute the fraton of all generated Rng Rng Rng Rng Rng number Fgure 6: The normalzed average fraton of alls beng routed to a node wth nreasng dstane from the enter. The arrval rate s alls per seond at a node.. The results ndate that an deal load-balanng based routng protool an ome lose to the theoretal upper bound. alls that arrve at a node. We ompute the average of ths fraton for all nodes that belong to a rng and hene an be onsdered to be at a fxed dstane from the enter of the network. The Fgures 5 and 6 plot ths average fraton (normalzed so that the least number s and all the others are dvded by ths least number) for two dfferent loads on the network. In both ases, shortest-path routng has a hgh load near the enter. PORS shfts ths load to the perphery but nurs the ost of hgher path length. W behaves lke shortest-path when the network s lghtly loaded but shfts the alls to the perphery wth an nreasng load. The dfferene between the theoretal upper bound and the expermental results s partly the result of the approxmatons and assumptons used n our model. However, the dfferene also reflets the nadequay of the exstng protools n load-balanng. The nrease n the all aeptane probablty of the load - balanng shemes as ompared to shortest-path routng ndates the mportane of load-balanng n ensurng better throughput n terms of all aeptane. In fat, load-balanng seems to be an mportant method of approahng P Max A 78

11 Saturaton Probablty Class Class Saturaton Probablty vs Saturaton Probablty Fgure 7: Varaton of Saturaton Probablty vs. Determnst Guarantee Lmt PA = PA = PA_Class <= Fgure 8: Rank-based prorty sheme. PA = PA_Class < PA_Class 5. System Saturaton Probablty The varaton of the probablty of system saturaton wth load s shown n Fgure 7. Ths metr remans near zero for moderate-toheavy loads, and takes on an appreable value only at hgh values of load. Ths ndates that system saturaton s a rare ourrene for the ommon values of load. Thus, the network rarely enters a state where every new all s rejeted. Ths also mples that for the ommon values of load, t s always possble to ensure that some fraton of the alls are guaranteed aeptane. Ths fraton s based on the values of the probablty of all aeptane at that load. 5. Determnst Guarantees Our am s to ensure that a ertan number of alls n the network an be assured of aeptane. We an do so by peggng these alls at a hgh prorty. Consder the followng rank-based prorty sheme: (Fgure 8) Calls are prortzed aordng to the lasses to whh they belong. In addton, alls that belong to the hghest prorty are further alloated to sub-lasses whh are based on the address or ID of the soure of the all. Further, all admsson ensures that only one all of a gven sub-lass exsts n the system. Ths mples that a partular node an orgnate only one suh hghest prorty all. Preempton s permtted amongst the sub-lasses themselves so that a hgh-prorty sub-lass has a better hane of aeptane. Hene a senaro an be envsaged as follows: the network s deployed n a mltary senaro n whh the nodes are under the ontrol of varous ommunatng offers. The node ID an be assgned based on the rank of the offer usng the node. Calls are prortzed at the tme of all admsson nto varous lasses. These alls then have probabltes of aeptane dependng on the lass to whh they have been assgned and the network state. In addton, the alls of the hghest prorty lass are assgned to sub-lasses based on ther node ID. Thus, to ensure that the all of the hghest-rankng offer (say the General) always gets through, the general s node would be assgned a hgh-prorty node ID. Thus, a set of nodes an be desgnated to ensure ertan all aeptane. To ensure that these guarantees provded are effetve, we need to estmate the number of alls (whh s equvalent to the number of sub-lasses) for whh ertan all aeptane an be ensured, and the all aeptane for the sub-lasses whh le outsde the former lass. 5.. Determnst Guarantee Lmt The Determnst guarantee lmt D refers to the number of sublasses of the hghest prorty lass that an be ensured determnst all aeptane as outlned at the begnnng of ths seton. These sub-lasses are referred to as the determnst sub-lasses. From Number of alls n R(j) = x Number of used slots n R(j) x, (4) f x =, then the number of used slots n R(j). If the total number of sub-lasses n the network =, then for every node j, the number of used slots n R(j). Thus, ths s the number of sub-lasses that an be defntely aepted by every regon of the network at a gven tme. y alloatng a unque set of slots to eah of the sub-lasses, we an ensure that alls of these sublasses are aepted (of ourse, any lower prorty alls may need to be preempted n the proess). Thus, the Determnst guarantee lmt D. Ths mples that sub-lasses an be ensured determnst all aeptane. However, ths beng a lower bound t may be possble for some more sub-lasses to be ensured of ths determnst aeptane. Independene of the Guarantee Lmt and Moblty At ths pont, we also would lke to pont out the effet of the moblty of the nodes on the lmt. The determnst guarantee lmt s ndependent of the moblty. The set of sub-lasses {,, } are ensured of determnst aeptane even n the fae of node moblty. Moblty n the network leads to path breaks and, subsequent, route reonfguraton attempts. In any suh attempt, the alls belongng to the determnst sub-lasses retan ther prorty. Thus, these alls are guaranteed resoures durng the reonfguraton. 5.. Probablty of Aeptane for the Probablst Sub-lasses The sub-lasses other than the determnst sub-lasses are referred to as the probablst sub-lasses. Sne sub-lasses are assgned based on node IDs, there are N sub-lasses, desgnated {,,N} n dereasng order of prorty. We are onsderng the all aeptane of a all belongng to a sub-lass n> (all } are of a hgher pror- alls n any of the sub-lasses {,, ty than ths all and are wthn the determnst guarantee lmt) at a tme t. We denote the probablty that a all of sub-lass exsts n the network at tme t by p (t). Let q (t) = p (t). Denote: the aeptane of all of sub-lass n as ACC n, and the number of alls sub-lasses {,,n } as Count(,n ). As n Equaton (4), f Count(,n ) s less than, then for 79

12 every node j, the number of slots used by alls of these sub-lasses s. All the remanng slots are ether free or are used by lower-prorty alls whh an be preempted by the all belongng to sub-lass n. Thus, the all of sub-lass n an be aepted. Thus At tme t, Count(,n ) < Call of sub lass n s aepted P (Call of sub lass n s aepted Count(,n ) < )= (5) y denotng the probablty of aeptane of the all belongng to sub-lass n at tme t as P n(t): P n(t) = [P (ACC n Count(,n ) < P (Count(,n ) < )] + [P (ACC n Count(,n ) P (Count(,n ) P n(t) P (Count(,n ) < ) P n(t) p l (t) S {,,n } l S j k S r {,,n } S ) ) )] (6) q r(t) (7) When the alls at eah node follow an dental probablty dstrbuton.e., p j(t) =p(t), j {,,N}, Equaton (7) smplfes to P n(t) j k = n p(t) q(t) n (8) = 6. CONCLUSION A realst analyss of the nature of QoS guarantees s rual n the desgn of new protools and the mprovement of exstng ones to handle the growng dversty of demands on networks. In ths paper, we have analyzed a TDMA-based Ad ho wreless network. We have derved an upper bound on the probablty of all aeptane: a bound that gves us a measure of the number of alls that an be allowed nto the network, and a lower bound on the probablty of system saturaton: a number that ndates the lkelhood of the network beng unable to aept any further alls. Our analyss takes nto onsderaton the behavor of the routng protool and the nter-dependene of resoures (tme-slots) of neghborng regons n a wreless network. Further, our smulaton studes ndate that the set of protools tested fall short of the establshed bounds. Amongst the three protools ompared, the one that norporated load-balanng out-performed the shortest-path routng based protool. Ths learly ndates the mportane of load-balanng n the attanment of hgh network performane, and the provson of better QoS guarantees. We have estmated the determnst guarantee lmt. Ths lmt ndates that t s always possble to ensure QoS guarantees for a ertan sub-lass of alls rrespetve of the moblty and resoure onstrants of the network. When the nodes are movng, the number of nodes n a gven regon beomes tme-dependent. Ths n turn s refleted n the fator beomng tme-ndependent. We are studyng the effet of tme-dependene of on the all aeptane. The expermental studes n ths work were performed wth a sngle-lass of alls. The next step would nvolve studyng the effet of ntrodung multple lasses of alls. Further, we are workng on extendng the analyss to handle the ase of all preempton, and on obtanng tghter estmates. The modelng of the routng algorthm needs to be refned so that we an make predtons based on the f k (, j) of dfferent protools. The expermental studes also need to be extended to ompare other protools to nfer the essental and desrable propertes of protools that approah optmalbehavor. 7. REFERENCES [] P. Gupta and P. R. umar, The Capaty of Wreless Networks, IEEE Transatons on Informaton Theory, vol. 46, no., pp , Marh. [] P. Gupta and P. R. umar, Towards an Informaton Theory of Large Networks: An Ahevable Rate Regon, IEEE Transatons on Informaton Theory, vol 49, no. 8, pp , August. [] M. Grossglauser and D. N. C. Tse, Moblty Inreases the Capaty of Ad ho Wreless Networks, IEEE/ACM Transatons on Networkng, vol., no. 4, pp , August. [4] U. C. ozat and L. Tassulas, Throughput apaty of Random Ad ho Networks wth Infrastruture Support, n proeedngs of ACM MOICOM, pp , September. [5] T. heemarjuna Reddy, I. arthgeyan,. S. Manoj, and C. Sva Ram Murthy, Qualty of Serve Provsonng n Ad ho Wreless Networks: A Survey of Issues and Solutons, to appear n Ad Ho Networks Journal. [6] P. Snha, R. Svakumar, and V. harghavan, CEDAR: A Core-Extraton Dstrbuted Ad ho Routng Algorthm, IEEE Journal on Seleted Areas n Communatons, vol. 7, no. 8, pp , August 999. [7] W. Lao,. Tseng, and. Shh, A TDMA-based andwdth Reservaton Protool for QoS Routng n a Wreless Moble Ad ho Network, n proeedngs of IEEE ICC, vol. 5, pp. 86-9, May. [8] G. arua and I. Chakraborty, Adaptve Routng for Ad ho Wreless Networks Provdng QoS Guarantees, n proeedngs of IEEE ICPWC, pp. 96-, Deember. [9] John G. emeny and J. L. Snell, Fnte Markov Chans,Van Nostrand, New ork, 96. [] P. uhholz, G. Cardo, P. emper, and S. Donatell, Complexty of Memory-effent roneker Operatons wth Applatons to the Soluton of Markov Models, INFORMS Journal on Computng, vol., no., pp. -, July. [] D. ertsekas and R. Gallager, Data Networks, Prente-Hall, New Jersey, 99. [] G. haya,. S. Manoj, and C. Sva Ram Murthy, Rng-ased Routng Shemes for Load Dstrbuton and Throughput Improvement n Mult-hop Cellular, Ad ho, and Mesh Networks, n proeedngs of HPC, LNCS 9, pp. 5-6, Deember. [] Qngmng Ma, Peter Steenkste, and Hu Zhang, Routng Hgh-bandwdth Traff n Max-mn Far Share Networks, n proeedngs of ACM SIGCOMM, pp. 6-7, August